60-second revision
- Subset → circle inside circle; no common members → separate circles; some common → overlapping circles.
- Check every pair of the three groups separately, then combine.
- Two sets: n(A or B) = n(A) + n(B) − n(A and B).
- Three sets: n(A or B or C) = A + B + C − (AB + BC + CA) + ABC.
- Neither = Total − n(at least one).
- Only A = n(A) − n(A and B) for two sets.
- Fill regions from the centre outwards: all-three first, then pairs, then the "only" regions.
Part 1: choosing the diagram
For "which diagram best represents" questions, look at each pair of groups and decide one of three relations. Then combine the three answers into one picture.
| Relation between two groups | Picture | Example |
|---|---|---|
| One is wholly inside the other | Circle inside circle | Rose, Flower |
| No member in common | Separate circles | Rose, Lotus |
| Some members in common | Overlapping circles | Singers, Teachers |
The standard three-group pictures
| Groups | Diagram |
|---|---|
| Kerala, India, World | Three concentric circles (each inside the next) |
| Fruits, Mango, Banana | Two separate circles inside a bigger circle |
| Women, Mothers, Engineers | Mothers inside Women; Engineers overlaps both |
| Doctors, Singers, Women | Three mutually overlapping circles |
| Cows, Mammals, Fish | Cows inside Mammals; Fish separate |
| Pens, Books, Chairs | Three separate circles |
| Odd numbers, Prime numbers, Integers | Two overlapping circles inside a bigger circle |
Part 2: counting with two sets
When two groups overlap, simply adding them counts the overlap twice. Hence n(A or B) = n(A) + n(B) − n(A and B). Everything else follows from it.
| Want | Formula |
|---|---|
| At least one | A + B − Both |
| Both | A + B − (at least one) |
| Only A | A − Both |
| Exactly one | (A − Both) + (B − Both) |
| Neither | Total − (at least one) |
Worked example: of 120 workers, 70 know Excel, 55 know Tally and 15 know neither. At least one = 120 − 15 = 105. Both = 70 + 55 − 105 = 20. Only Excel = 50, only Tally = 35. Check: 50 + 20 + 35 + 15 = 120.
Part 3: counting with three sets
Use n(A or B or C) = A + B + C − AB − BC − CA + ABC, where AB means "in both A and B" (including those in all three). For region-wise questions, fill a diagram from the centre: write ABC, then each pair region minus ABC, then each "only" region.
Worked example: 70 like rice (R), 60 roti (T), 50 dosa (D); RT 25, TD 20, RD 15, all three 10; total surveyed 150. At least one = 70 + 60 + 50 − 25 − 20 − 15 + 10 = 130, so 20 like none. Only rice = 70 − (25 − 10) − (15 − 10) − 10 = 70 − 15 − 5 − 10 = 40.
Words that change the region
| Wording | Region |
|---|---|
| At least one | Everything inside any circle |
| Only A / A alone | Inside A, outside all other circles |
| Exactly two (three sets) | Sum of the three pair-only regions |
| At most one | The three only-regions plus none |
| Neither / none | Outside every circle |
Worked example with three sets: if all three = 4 and the pair figures are AB 10, BC 8 and CA 6, then exactly two = (10 − 4) + (8 − 4) + (6 − 4) = 12. Adding 10 + 8 + 6 = 24 would count the all-three group three times.
Part 4: reading a described diagram
Some questions describe regions with numbers ("inside A only 14, overlap 6, inside B only 9"). The total in A is A-only plus every overlap that lies inside A. Read each circle as everything it encloses, not just its "only" part.
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